|King Fahd University of
Petroleum and Minerals
Department of Mathematics and Statistics
Fractional Models in Science & Engineering
Theory and Computation
December 10, 2018
KFUPM Conference Room (Building # 20)
Department of Mathematical Sciences &
Middle Tennessee State University, USA.
A split-step second order predictor-corrector method
for space-fractional reaction-diffusion equations with nonhomogeneous
boundary conditions is presented and Matrix Transfer Technique (MTT) is used
for spatial discretization. The method is shown to be unconditionally stable
and second order convergent. Numerical experiments are performed to confirm
the stability and second order convergence of the method. The split-step
predictor-corrector method is also compared with an IMEX predictor-corrector
method which is found to incur oscillatory behavior for some time steps. Our
method is seen to produce reliable and oscillation free results for any time
step when implemented on numerical examples with no smooth initial data. We
also present a priori reliability constraint for IMEX predictor-corrector
method to avoid unwanted oscillations and show its validity numerically on
several test problems from the literature.
Department of Mathematics
University of Bari, Italy.
On special fractional operators in
computational electromagnetism: theory and numerical methods.
The propagation of electric and magnetic fields
in complex systems, such as for instance special polymers or biologic
tissues, is usually described in the frequency domains by means of functions
with more than one real powers. The characterization of these models in the
time domain involves new non-standard differential operators of fractional
order . In this talk we discuss the constitutive law of the
Havriliak-Negami model which allows to close the Maxwell's system and
describe the interactions between electric and magnetic fields in a wide
range of materials, with important applications in engineering and medicine.
describing new recent developments for characterizing in the time domain the
operators associated with this model, we focus with the connections with
fractional integrals and fractional derivatives based on the so-called
Prabhakar function  and we discuss some of the possible approaches for
the discretization of these operators and for their use in numerical
 R. Garrappa, F. Mainardi, G.
Maione, Models of dielectric relaxation based on completely monotone
Fract. Calc. Appl.
Anal., 2016, 19(5), 1105-1160
 R. Garra, R. Garrappa, The
Prabhakar or three parameter Mittag--Leffler function: theory and
Commun. Nonlinear Sci. Numer.
Simul., 2018, 56, 314-329
 R. Garrappa, On Grunwald-Letnikov operators for fractional relaxation in
Havriliak-Negami models, Commun.
Simul., 2016, 38, 178-191
Department of Electrical & Computer
University of Porto, Portugal.
Fractional Calculus: The Perspective of Complex Systems.
Fractional Calculus (FC) started in 1695 when L'HŰpital wrote a letter to
Leibniz asking for the meaning of Dny for n = 1/2. Starting with the ideas of
Leibniz many important mathematicians developed the theoretical concepts. By the
beginning of the twentieth century Olivier Heaviside applied FC in electrical
engineering, but his visionary and important contributions were forgotten during
several decades. Only in the eighties FC emerged associated with phenomena such
as fractal and chaos and, consequently, in nonlinear dynamical. In the last
years, FC become a 'new' mechanism for the analysis of dynamical systems. FC is
now recognized to be an important tool to model and control systems with long
range memory effects. This lecture introduces several applications in distinct
areas of science and engineering based on the authorís own experience and
research work during the last years. The presentation focusses advanced topics,
out of the standard stream, usually followed by the scientific community, where
the FC and complex systems methods reveal important properties.
Department of Mathematics
COMSATS University, Pakistan.
Problems for Some Space-Time Fractional Differential Equations.
Fractional Calculus (FC), that is, the study of
arbitrary order integrals and derivatives has beenconsidered by many experts
from all fields of sciences. FC has applications in many fields just
tomention a few are in biology, physics, finance, viscoelasticity processes,
prediction of extreme eventslike earthquake etc. The fractional derivatives
in time and space have been considered to explain theanomalies in the
complex phenomena of diffusion/transport at both micro and macro level. The
timefractional diffusion equations with Riemann-Liouville or Caputo
fractional derivatives are equivalentto infinitesimal generators of time
fractional evolutions that arise in the transition from microscopicto
macroscopic time scales. The transition from first order derivative to the
fractional order timederivative arise physically reported by many see for
Inverse problems for fractional differential equations have been considered
by many in the recentpast which include inverse source problems, inverse
coefficient problems etc. Indeed, the inverse problemsfor time, space and
time-space fractional diffusion equations are considered. In this talk weare
going to discuss the inverse source problems for the Space-Time Fractional
Differential Equations(STFDEs). The STFDE is obtained from classical
diffusion equation by replacing time derivativewith fractional order time
derivative and Sturm-Liouville operator by fractional order
Sturm-Liouvilleoperator. The spectral problem of the STFDEs is the
fractional order Sturm-Liouville system, whichis a generalization of the
well known Sturm-Liouville system involving fractional derivatives in
spacedefined in Caputoís sense with Dirichlet boundary conditions. By
variational techniques several properties of eigenvalues and eigenfunctions
of the fractional order Sturm-Liouville system were investigated in .
In the first part of my talk, an inverse
problem of recovering a space dependent source term along with solution for
a STFDE from over-specified condition of final data will be discussed .
The conditions on the given data such that a
unique solution of the inverse problem exists will be presented. Secondly,
inverse problem of determining a time dependent source term from the total
energy measurement of the system (the over-specified condition) for a STFDE
will be considered .
existence and uniqueness results are proved by using eigenfunction expansion
method. Several special cases, particular examples will be provided. Some
future perspectives will be shared with the audience.
- R. Hilfer, Applications of Fractional Calculus in
Physics, World Scienti c, 2000.
- M. Klimek, A.B.
Malinowska, T. Odzijewicz, Variational methods for the fractional
problem, J. Math. Anal. Appl., 416, (2014), 402-428.
- M. Ali, S. Aziz and S.A.
Malik, Inverse source problem for a space-time fractional diffusion
Calc. Appl. Anal., 21 (2018), 844-863.
- M. Ali, S. Aziz, S.A.
Malik, Inverse problem for a space-time fractional diffusion equation:
fractional Sturm-Liouville operator, Mathematical Methods in the Applied
Sciences, 41 (2018),
Saudi Aramco Expec ARC Computational Modeling Technology (CMT) Team
Fracture modeling in the current reservoir simulation applications and potential for fractional model
Numerical simulation of
fractured reservoirs is a challenging task due to the huge difference
between the scales of velocity in the fractures and the rock matrix. In
fractured reservoirs, the matrix contributes most of the fluid storage but
very little on fluid transport; on the contrary, the fracture system
provides high flow capacity but low pore volume . A common approach in
dealing with this problem is applying dual or multi-porosity models, which
is based on separation of scales. This talk will be about the current state
of art in the applied fractured reservoir simulation and explore where the
fractional diffusion equations would fit and make difference for the future
Bicheng Yan, Masoud Alfi, Cheng An, Yang Cao, Yuhe Wang, John E. Killough,
General Multi-Porosity simulation for fractured reservoir modeling, Journal
of Natural Gas Science and Engineering, Volume 33, 2016, Pages 777-791